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We consider a reductive algebraic group G defined over the reals which
acts linearly on a vector space V . It is classically known that the algebra of invariant
polynomials is finitely generated, which allows one to construct an algebraic orbit
map V − > V/G. After introducing the setup and recalling some important results
by Procesi-Schwarz, Br¨ocker and others, we will take a closer look at G-invariant
polynomials which are positive or sums of squares.
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